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ARITHMETIC THE SECOND
product of (flls az) and C61S &2) is (a^b^ a^b^ azbit ajbz). It is
always possible to reduce any such product-symbol (for a field
of degree n) to a symbol containing at most n integers.
One final short remark completes the synopsis of the story.
An ideal whose symbol contains but one integer, such as (GX), is
called a principal ideal. Using as before the notation (a-^ ' (b^)
to signify that (a-^ contains (b^, we can see without difficulty
that (ax) ! (&i) when, and only when, the integer 0T divides the
integer br As before, then, the concept of arithmetical divisi-
bility is here - for algebraic integers - completely equivalent to
that of class inclusion. A prime ideal is one which is not 'divi-
sible by* - included in - any ideal except the all-inclusive ideal
which consists of all the algebraic integers in the given field.
Algebraic integers being now replaced by their corresponding
principal ideals, it is proved that a given ideal is a product of
prime ideals hi one way only, precisely as in the 'fundamental
theorem of arithmetic' a rational integer is the product of
primes in one way only. By the above equivalence of arith-
metical divisibility for algebraic integers and class inclusion,
the fundamental theorem of arithmetic has been restored to
integers in algebraic number fields.
Anyone who will ponder a little on the foregoing bare outline
of Dedekmd's creation will see that what he did demanded
penetrating insight and a mind gifted far above the ordinary
good mathematical mind in the power of abstraction. Dedekind
was a mathematician after Gauss' own heart: 'At nostro quidem
juditio hujusmodi veritates ex notiombus polios quam ex nota-
tionibus hauriri debeanf (But in our opinion such truths
•[arithmetical] should be derived from notions rather than from
notations). Dedekind always relied on his head rather than on
an ingenious symbolism and expert manipulations of formulae
to get hi™ forward. If ever a man put notions into mathematics,
Dedekind did, and the wisdom of his preference for creative
ideas over sterile symbols is now apparent although it may not
have been during his lifetime. The longer mathematics lives tlie
more abstract - and therefore, possibly, also the more practical
- it becomes.